The Extremal Graphs for (Sum-) Balaban Index of Spiro and Polyphenyl Hexagonal Chains

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1 Iranian J. Math. Chem. 9 (4) December (2018) Iranian Journal of Mathematical Chemistry Journal homepage: ijmc.kashanu.ac.ir The Extremal Graphs for (Sum-) Balaban Index of Spiro and Polyphenyl Hexagonal Chains YANG ZUO, YAQIAN TANG AND HANYUAN DENG College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan , P. R. China ARTICLE INFO Article History: Received 9 August 2018 Accepted 1 October 2018 Published online 15 December 2018 Academic Editor: Sandi Klavžar Keywords: Balaban index Sum-Balaban index Spiro hexagonal chain Polyphenyl hexagonal chain ABSTRACT As highly discriminant distance-based topological indices, the Balaban index and the sum-balaban index of a graph G are defined as J(G) = and SJ(G) = () () () (), respectively, where D (u) = d(u, v) is the distance sum of a vertex u in G, m is the number of edges and μ is the cyclomatic number of G. They are useful distance-based descriptor in chemometrics. In this paper, we focus on the extremal graphs of spiro and polyphenyl hexagonal chains with respect to the Balaban index and the sum-balaban index University of Kashan Press. All rights reserved 1 INTRODUCTION Polyphenyl and spiro hexagonal chains have been widely investigated, and they represent a relevant area of interest in mathematical chemistry because they have been used to study intrinsic properties of molecular graphs. Polyphenyls and their derivatives, which can be used in organic synthesis, drug synthesis, heat exchangers, etc., attracted the attention of chemists for many years [7, 8, 20, 21, 26, 28, 30]. Spiro compounds are an important class of cycloalkanes in organic chemistry. A spiro union in spiro compounds is a linkage between two rings that consists of a single atom common to both rings and a free spiro union is a linkage that consists of the only direct union between the rings. Several works have been developed to analyze extremal values and extremal graphs for many topological indices on the spiro and polyphenyl hexagonal chains. Some results on energy, Merrifield- Simmons index, Hosoya index, Wiener index and Kirchhoff index of the spiro and Corresponding Author ( address: hydeng@hunnu.edu.cn) DOI: /ijmc

2 242 ZUO, TANG AND DENG polyphenyl chains were reported in [2, 9, 12, 13, 16, 17, 35, 32]. In this paper, we will consider the extremal values and the extremal graphs for the Balaban index and the sum- Balaban index on polyphenyl and spiro chains. As a highly discriminant distance-based topological index, the Balaban index [3] was defined on the basis of the Randić formula but using distance sums for vertices instead of vertex degrees. The Balaban index is a variant of connectivity index, represents extended connectivity and is a good descriptor for the shape of the molecules. It shows a good isomer discrimination ability and produces good correlations with some physical properties, such as the motor octane number [6], and it appears in theoretical models for predicting and screening drug candidates in rational drug design strategies [22]. It is of interest in combinatorial chemistry. It turned out to be applicable to several questions of molecular chemistry. Throughout this paper we consider only simple and connected graphs. For a graph G with vertex set V(G) and edge set E(G). The distance between vertices u and v in G, denoted by d (u, v), is the length of a shortest path connecting u and v. Let D (u) = () d(u, v), which is the distance sum of vertex u in G. The cyclomatic number μ of G is the minimum number of edges that must be removed from G in order to transform it to an acyclic graph. Let V(G) = n, E(G) = m, it is known that μ = m n 1. The Balaban index of a connected graph G is defined as J(G) = (). () () It was introduced by A. T. Balaban in [3, 4], which is also called the average distance-sum connectivity or J index. It appears to be a very useful molecular descriptor with attractive properties. In 2010, Balaban et al. [5] also proposed the sum-balaban index SJ(G) of a connected graph G, which is defined as SJ(G) = () () (). The Balaban index and the sum-balaban index were used in various quantitative structure-property relationship and quantitative structure activity relationship studies. Until now, the Balaban index and the sum-balaban index have gained much popularity and new results related to them are constantly being reported, see [1, 10, 11, 14, 15, 18, 19, 25, 27, 29, 31, 33, 34]. Let G be a cactus graph in which each block is either an edge or a hexagon. G is called a polyphenyl hexagonal chain if each hexagon of G has at most two cut-vertices, and each cut-vertex is shared by exactly one hexagon and one cut-edge. The number of hexagons in G is called the length of G. An example of a polyphenyl hexagonal chain is shown in Figure 1.

3 The Extremal Graphs for (Sum-) Balaban Index of some Hexagonal Chains 243 Figure 1: A polyphenyl hexagonal chain of length 8. Let PPC = H H H (n 3) be a polyphenyl hexagonal chain of length n. There is a cut-edge v u between PPC and H, see Figure 2. Note that any polyphenyl hexagonal chain of length n has 6n vertices and 7n 1 edges. A vertex v of H is said to be ortho-, meta-, and para-vertex if the distance between v and u is 1, 2 and 3, denoted by o, m and p, respectively. Example of Figure 2, o = x, x, m = x, x, p = x. Obviously, every hexagon has two ortho-vertices, two meta-vertices and one para-vertex except the first hexagon H. A polyphenyl hexagonal chain PPC is a polyphenyl ortho-chain if v = o for 2 k n 1. The polyphenyl meta-chain and polyphenyl para-chain are defined in a completely analogous manner. Figure 2: A polyphenyl hexagonal chain of length n. The polyphenyl ortho-, meta-, and para-chains of length n are denoted by O, M and P, respectively. Examples of polyphenyl ortho-, meta-, and para-chains are shown in Figure 3. Figure 3: Polyphenyl hexagonal ortho-, meta-, and para-chains of length 7.

4 244 ZUO, TANG AND DENG The definition of spiro hexagonal chain is same as definition of polyphenyl hexagonal chain. A hexagonal cactus is a connected graph in which every block is a hexagon. A vertex shared by two or more hexagon is called a cut-vertex. If each hexagon of a hexagonal cactus G has at most two cut-vertices, and each cut-vertex is shared by exactly two hexagons, then G is called a spiro hexagonal chain. The number of hexagon in G is called the length of G. An example of a spiro hexagonal chain is shown in Figure 4. Figure 4: A spiro hexagonal chain of length 7. Obviously, a spiro hexagonal chain of length n has 5n 1 vertices and 6n edges. Let SPC = H H H (n 3) be a spiro hexagonal chain of length n. There is a cutvertex u between SPC and H, see Figure 5. Figure 5: A spiro hexagonal chain of length n. A vertex v of H is said to be ortho-, meta-, and para- vertex if the distance between v and u is 1, 2 and 3, denoted by o, m and p, respectively. A spiro hexagonal chain is a spiro ortho-chain if u = o for 2 k n. The spiro meta-chain and para-chains are defined in a completely analogous manner. The spiro ortho-, meta-, and para-chains of length n are denoted by SO, SM and SP, respectively. The following lemmas will be used in the next section. Lemma 1 ([14]) Let x, y, a R such that x y a. Then equality if and only if x = y a. ()() with Lemma 2 ([15]) Let r, t, r, t R such that r > t and r r = t t > 0. Then <. Lemma 3 ([14]) Let a, w, x, y, z R such that,. Then

5 The Extremal Graphs for (Sum-) Balaban Index of some Hexagonal Chains 245 ()() ()(). 2. (SUM-) BALABAN INDEX OF POLYPHENYL HEXAGONAL CHAINS In this section, we first give two cut-edge transformations on PPC, and then determine the extremal graphs by using the transformations. The first cut-edge transformation on PPC n : Let G = H H H (n 3) be a polyphenyl hexagonal chain of length n. x and x are two cut-vertices in the k th hexagon H, and the distance between x and x is 3. If G is the graph obtained from G by deleting the cut edge x u between H and H, and adding a new cut-edge x u between H and H (see Figure 6), then we say that G is obtained from G by the first cut-edge transformation. Figure 6: The first cut-edge transformation. Lemma 4 Let G = H H H (n 3) be a polyphenyl hexagonal chain of length n. G is obtained from G by the first cut-edge transformation. Then J(G) < J(G ) and SJ(G) < SJ(G ). Proof. Let F = H H H, F = H, F = H H H. The length of F is a = k 1 and the length of F is b = n k. Obviously, a b = n 1. Without loss of generality, let a b. For a vertex v F, we have D (v ) = d (v, u) d (v, u) d (v, u), D (v ) = d (v, u) d (v, u) d (v, u) d (v, u) = d (v, u), d (v, u) = d (v, u), d (v, u) = d (v, u) 6b. So, D (v ) D (v ) = 6b and D (v ) > D (v ). For a vertex v F, we have D (v ) = d (v, u) d (v, u) d (v, u), D (v ) = d (v, u) d (v, u) d (v, u).

6 246 ZUO, TANG AND DENG Furthermore, d (v, u) = d (v, u), d (v, u) = d (v, u), d (v, u) = d (v, u) 6a. So, D (v ) D (v ) = 6a D (v ) > D (v ). For a vertex in V(F ) = {x, x, x, x, x, x }, it is easy to see that D (x ) D (x ) = D (x ) D (x ) = D (x ) D (x ) = 6b, D (x ) D (x ) = D (x ) D (x ) = D (x ) D (x ) = 6b. (I) For an edge v v E(F ) E(F ), we have and ( ) ( ) > ( ) ( ) > ( ) ( ) ( ) ( ) since D (v ) > D (v ) and D (v ) > D (v ). (II) In what follows, we consider an edge in {x x, x x, x x, x x, x x, x x, x v, x u }. Let M = d (x, u) d (x, u) d (x, u), where x {x, x, x, x, x, x }. Then M = d (x, u) d (x, u) d (x, u). It can be checked directly that D (x ) = M 18bD (x ) = M 12b D (x ) = M 6a 12bD (x ) = M 6a 6b D (x ) = M 12a 6bD (x ) = M 12a D (x ) = M 18aD (x ) = M 18a 6b D (x ) = M 12a 6bD (x ) = M 12a 12b D (x ) = M 6a 12bD (x ) = M 6a 18b. (i) For the edges x v, x u E(G) and x v, x u E(G ), we have and ( ) ( ) ( ) ( ) > > ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1) (2) (3) ( ) ( ). (4) since D (x ) > D (x ), D (v ) > D (v ), D (x ) > D (x ), D (u ) > D (u ). (ii) For the edges x x, x x E(G), we have D (x ) D (x ) 6b, D (x ) = D (x ) 6b and D (x ) = D (x ) 6b. By Lemma 1, we can get and ( ) ( ) ( ) ( ) (5)

7 The Extremal Graphs for (Sum-) Balaban Index of some Hexagonal Chains 247 ( ) ( ) ( ) ( ) Also, D (x ) D (x ) 6b, D (x ) = D (x ) 6b and D (x ) = D (x ) 6b, by Lemma 1, we have and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (iii) For the edges x x, x x E(G), let x = D (x ), y = D (x ), w = D (x ), z = D (x ). Then D (x ) = x 6b, D (x ) = y 6b, D (x ) = w 6b, D (x ) = z 6b. Note that w > x, z > y and >, >, by Lemma 3, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Now, let r = D (x ) D (x ) = 2M 30a 6b, r = D (x ) D (x ) = 2M 30a 18b, t = D (x ) D (x ) = 2M 6a 18b, t = D (x ) D (x ) = 2M 6a 30b. Then r r = t t = 12b > 0, r t = 24a 12b > 0 (since a b > 0). By Lemma 2, we have > ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (iv) For the edges x x, x x E(G), by the same ways as in (iii), we can get ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) From Equations (1 12) and the definition of the Balaban index and the sum- Balaban index, we have J(G) < J(G ) and SJ(G) < SJ(G ). The second cut-edge transformation on PPC n : Let G = H H H (n 3) be a polyphenyl hexagonal chain of length n. x and x are two cut-vertices in the k th hexagon H, and the distance between x and x is 2. If G is the graph obtained from G by deleting the cut edge x u between H and H, and adding a new cut-edge x u between H and H (see Figure 7), then we say that G is obtained from G by the second cut-edge transformation. Lemma 5 Let G = H H H (n 3) be a polyphenyl hexagonal chain of length n. G is obtained from G by the second cut-edge transformation. Then J(G) < J(G ) and SJ(G) < SJ(G ). (6) (7) (8) (9) (10) (11) (12)

8 248 ZUO, TANG AND DENG Proof. Let F = H H H, F = H, F = H H H. The length of F is a = k 1 and the length of F is b = n k. Obviously, a b = n 1. Without loss of generality, let a b. Figure 7: The second cut-edge transformation. For a vertex v F, we have D (v ) = d (v, u) d (v, u) d (v, u), D (v ) = d (v, u) d (v, u) d (v, u) and d (v, u) = d (v, u), d (v, u) = d (v, u), d (v, u) = d (v, u) 6b. So, D (v ) D (v ) = 6b and D (v ) > D (v ). For a vertex v F, we have D (v ) = d (v, u) d (v, u) d (v, u), D (v ) = d (v, u) d (v, u) d (v, u) and d (v, u) = d (v, u), d (v, u) = d (v, u), d (v, u) = d (v, u) 6a. So, D (v ) D (v ) = 6a and D (v ) > D (v ). For a vertex in F = {x, x, x, x, x, x }, let M = d (x, u) d (x, u) d (x, u) = d (x, u) d (x, u) d (x, u), where x {x, x, x, x, x, x }. It can be checked directly that D (x ) = M 12bD (x ) = M 6b D (x ) = M 6a 6bD (x ) = M 6a D (x ) = M 12aD (x ) = M 12a 6b D (x ) = M 18a 6bD (x ) = M 18a 12b D (x ) = M 12a 12bD (x ) = M 12a 18b D (x ) = M 6a 18bD (x ) = M 6a 12b. (I) For an edge v v E(F ) E(F ), we have D (v ) > D (v ), D (v ) > D (v ). So,

9 The Extremal Graphs for (Sum-) Balaban Index of some Hexagonal Chains 249 and ( ) ( ) > ( ) ( ) > ( ) ( ) ( ) ( ) (II) In what follows, we consider an edge in {x x, x x, x x, x x, x x, x x, x v, x u }. (i) For the edges x v, x u E(G) and x v, x u E(G ), it is easy to know that D (x ) > D (x ), D (v ) > D (v ), D (x ) > D (x ), D (u ) > D (u ). And ( ) ( ) > ( ) ( ) ( ) ( ) > ( ) ( ) ( ) ( ) ( ) ( ) (13) (14) ( ) ( ), (15) ( ) ( ). (16) (ii) For the edges x x, x x E(G), because D (x ) > D (x ) 6b, by Lemma 1, we have and ( ) ( ) ( ) ( ) = ( ) ( ) Also, because D (x ) = D (x ) 6b, by Lemma 1, we have and ( ) ( ) = ( ) ( ) (17) ( ) ( ). (18) ( ) ( ) (19) ( ) ( ). (20) (iii) For the edges x x, x x E(G), let x = D (x ), y = D (x ), w = D (x ), z = D (x ), then x 6b = D (x ), y 6b = D (x ), w 6b = D (x ), z 6b = D (x ). Note that w > x, z > y, >, >, by Lemma 3, we have ( ) ( ) ( ) ( ) > ( ) ( ) ( ) ( ). (21) Let r = D (x ) D (x ) = 2M 30a 6b, r = D (x ) D (x ) = 2M 30a 18b, t = D (x ) D (x ) = 2M 6a 6b, t = D (x ) D (x ) = 2M 6a 18b. Then r r = t t = 12b > 0, r t = 24a > 0. By Lemma 2, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (iv) For the edges x x, x x E(G), by the same way as in (iii), we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) > ( ) ( ) ( ) ( ) (22) ( ) ( ), (23) ( ) ( ). (24)

10 250 ZUO, TANG AND DENG From Equations (13 24) and the definitions of the Balaban index and the sum-balaban index, we have J(G) < J(G ) and SJ(G) < SJ(G ). Using the transformations above, we can get the extremal graphs for the (sum-) Balaban index on polyphenyl hexagonal chains. Theorem 6 Let PPC be a polyphenyl hexagonal chain of length n. Then J(P ) J(PPC ) J(O ), SJ(P ) SJ(PPC ) SJ(O ), with equalities if and only if PPC = O, PPC = P, respectively. Proof. Suppose on the contrary that G = H H H (n 3), a polyphenyl hexagonal chain of length n, has the maximum (sum-) Balaban index, and G O. Then there is 1 < k < n such that the distance between two cut-vertices u and v, which belongs to the k-th hexagon H, is 2 or 3. Let G be the graph obtained from G by using the first or the second cut-edge transformation. By Lemmas 4 and 5, we have J(G) < J(G ) and SJ(G) < SJ(G ), a contradiction. So, O is the unique graph with the maximum (sum-) Balaban index. Similarly, we can show that P is the unique graph with the minimum (sum-) Balaban index. 3. (SUM-) BALABAN INDEX OF SPIRO HEXAGONAL CHAINS As in the last section, we first give two transformations on SPC. The first cut-vertex transformation on SPC n : Let G = H H H (n 3) be a spiro hexagonal chain of length n, v = x and v = x are two cut-vertices in k-th hexagon H. If G is the graph obtained from G by taking two cut-vertices v = x and v = x in k -th hexagon H, then we say that G is obtained from G by the first cut-vertex transformation, see Figure 8. Figure 8: The first cut-vertex transformation.

11 The Extremal Graphs for (Sum-) Balaban Index of some Hexagonal Chains 251 Lemma 7 Let G = H H H (n 3) be a spiro hexagonal chain of length n. G is obtained from G by the first cut-vertex transformation. Then J(G) < J(G ) and SJ(G) < SJ(G ). Proof. Let F = H H H, F = H, F = H H H in Figure 8. V(F ) = {x, x, x, x, x, x } and the length of F and F is a and b, respectively, a b = n 1. Let M = d (x, u) d (x, u) d (x, u), where x V(F ). Then M = d (x, u) d (x, u) d (x, u). For a vertex v F, we have D (v ) = d (v, u) d (v, u) d (v, u), D (v ) = d (v, u) d (v, u) d (v, u), and d (v, u) = d (v, u), d (v, u) = d (v, u), d (v, u) = d (v, u) 6b. So, D (v ) D (v ) = 6b and D (v ) > D (v ). Similarly, we have D (v ) D (v ) = 6a for a vertex v F. For a vertex in V(F ) = {x, x, x, x, x, x }, it can be check directly that D (x ) = M 18b, D (x ) = M 12b D (x ) = M 6a 12b, D (x ) = M 6a 6b D (x ) = M 12a 6b, D (x ) = M 12a D (x ) = M 18a, D (x ) = M 18a 6b D (x ) = M 12a 6b, D (x ) = M 12a 12b D (x ) = M 6a 12b, D (x ) = M 6a 18b. Using the method as in Lemma 4, we can get Lemma 7. The second cut-vertex transformation on SPC n : Let G = H H H (n 3) be a spiro hexagonal chain of length n, v = x and v = x are two cut-vertices in k-th hexagon H. If G is the graph obtained from G by taking two cut-vertices v = x and v = x in k -th hexagon H, then we say that G is obtained from G by the second cut-vertex transformation (see Figure 9). Figure 9: The second cut-vertex transformation.

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